Weak convergence (Hilbert space)

In mathematics, weak convergence in a Hilbert space is convergence of a sequence of points in the weak topology.

Definition
A sequence of points $$(x_n)$$ in a Hilbert space H is said to converge weakly to a point x in H if


 * $$\lim_{n\to\infty}\langle x_n,y \rangle = \langle x,y \rangle$$

for all y in H. Here, $$\langle \cdot, \cdot \rangle$$ is understood to be the inner product on the Hilbert space. The notation


 * $$x_n \rightharpoonup x$$

is sometimes used to denote this kind of convergence.

Properties

 * If a sequence converges strongly (that is, if it converges in norm), then it converges weakly as well.
 * Since every closed and bounded set is weakly relatively compact (its closure in the weak topology is compact), every bounded sequence $$x_n$$ in a Hilbert space H contains a weakly convergent subsequence. Note that closed and bounded sets are not in general weakly compact in Hilbert spaces (consider the set consisting of an orthonormal basis in an infinite-dimensional Hilbert space which is closed and bounded but not weakly compact since it doesn't contain 0). However, bounded and weakly closed sets are weakly compact so as a consequence every convex bounded closed set is weakly compact.
 * As a consequence of the principle of uniform boundedness, every weakly convergent sequence is bounded.
 * The norm is (sequentially) weakly lower-semicontinuous: if $$x_n$$ converges weakly to x, then


 * $$\Vert x\Vert \le \liminf_{n\to\infty} \Vert x_n \Vert, $$


 * and this inequality is strict whenever the convergence is not strong. For example, infinite orthonormal sequences converge weakly to zero, as demonstrated below.


 * If $$x_n \to x$$ weakly and $$\lVert x_n \rVert \to \lVert x \rVert$$, then $$ x_n \to x$$ strongly:


 * $$\langle x - x_n, x - x_n \rangle = \langle x, x \rangle + \langle x_n, x_n \rangle - \langle x_n, x \rangle - \langle x, x_n \rangle \rightarrow 0.$$


 * If the Hilbert space is finite-dimensional, i.e. a Euclidean space, then weak and strong convergence are equivalent.

Example


The Hilbert space $$L^2[0, 2\pi]$$ is the space of the square-integrable functions on the interval $$[0, 2\pi]$$ equipped with the inner product defined by
 * $$\langle f,g \rangle = \int_0^{2\pi} f(x)\cdot g(x)\,dx,$$

(see Lp space). The sequence of functions $$f_1, f_2, \ldots$$ defined by
 * $$f_n(x) = \sin(n x)$$

converges weakly to the zero function in $$L^2[0, 2\pi]$$, as the integral
 * $$\int_0^{2\pi} \sin(n x)\cdot g(x)\,dx.$$

tends to zero for any square-integrable function $$g$$ on $$[0, 2\pi]$$ when $$n$$ goes to infinity, which is by Riemann–Lebesgue lemma, i.e.
 * $$\langle f_n,g \rangle \to \langle 0,g \rangle = 0.$$

Although $$f_n$$ has an increasing number of 0's in $$[0,2 \pi]$$ as $$n$$ goes to infinity, it is of course not equal to the zero function for any $$n$$. Note that $$f_n$$ does not converge to 0 in the $$L_\infty$$ or $$L_2$$ norms. This dissimilarity is one of the reasons why this type of convergence is considered to be "weak."

Weak convergence of orthonormal sequences
Consider a sequence $$e_n$$ which was constructed to be orthonormal, that is,


 * $$\langle e_n, e_m \rangle = \delta_{mn}$$

where $$\delta_{mn}$$ equals one if m = n and zero otherwise. We claim that if the sequence is infinite, then it converges weakly to zero. A simple proof is as follows. For x ∈ H, we have


 * $$ \sum_n | \langle e_n, x \rangle |^2 \leq \| x \|^2$$ (Bessel's inequality)

where equality holds when {en} is a Hilbert space basis. Therefore


 * $$ | \langle e_n, x \rangle |^2 \rightarrow 0$$ (since the series above converges, its corresponding sequence must go to zero)

i.e.


 * $$ \langle e_n, x \rangle \rightarrow 0 .$$

Banach–Saks theorem
The Banach–Saks theorem states that every bounded sequence $$x_n$$ contains a subsequence $$x_{n_k}$$ and a point x such that


 * $$\frac{1}{N}\sum_{k=1}^N x_{n_k}$$

converges strongly to x as N goes to infinity.

Generalizations
The definition of weak convergence can be extended to Banach spaces. A sequence of points $$(x_n)$$ in a Banach space B is said to converge weakly to a point x in B if $$f(x_n) \to f(x)$$ for any bounded linear functional $$f$$ defined on $$B$$, that is, for any $$f$$ in the dual space $$B'$$. If $$B$$ is an Lp space on $$\Omega$$ and $$p<+\infty$$, then any such $$f$$ has the form $$f(x) = \int_{\Omega} x\,y\,d\mu$$ for some $$y\in\,L^q(\Omega)$$, where $$\mu$$ is the measure on $$\Omega$$ and $$\frac{1}{p}+\frac{1}{q}=1$$ are conjugate indices.

In the case where $$B$$ is a Hilbert space, then, by the Riesz representation theorem, $$f(\cdot) = \langle \cdot,y \rangle$$ for some $$y$$ in $$B$$, so one obtains the Hilbert space definition of weak convergence.